Non-Geometric String Theory Backgrounds from Gauged Supergravities to Interacting Chiral Bosons ≺

Nikolaos Prezas

Albert Einstein Center for Fundamental PhysicsInstitute for Theoretical Physics

University of Bern

based on• "Conformal Chiral Boson Models on Twisted Doubled Tori and Non-Geometric

String Backgrounds", with S. Avramis and J. P. Derendinger,Nucl. Phys. B827 (2010) 281; arXiv:0910.0431

• "Worldsheet Theories for Non-Geometric String Backgrounds",with G. Dall’Agata, JHEP 0808 (2008) 088; arXiv:0712.1026

• "Gauged Supergravities from Twisted Doubled Tori and Non-Geometric StringBackgrounds", with G. Dall’Agata, H. Samtleben and M. Trigiante,Nucl. Phys. B799 (2008) 80; arXiv:0712.1026

Motivation and overview

Fluxes, twists and supergravity gaugings

Non-geometric backgrounds and twisted doubled tori

Interacting chiral boson models: Lorentz invariance

Interacting chiral boson models: one-loop effective action

Summary and open problems

Standard lore about string theory

I Consistent unification of gauge and gravitational interactions• However, it requires (typically) more dimensions than four!• How can we make contact to four-dimensional physics ?

Compactification approach

I Assumption: spacetime of the formM4 × Y6

• Classify possible internal spaces Y6

• Derive effective theory in 4 dimensions

Traditional compactification scheme (80s)

I Non-trivial metric on Y6 + SUSY → Calabi–Yau(or manifolds with exceptional holonomy)• However: huge vacuum degeneracy• Moduli: scalars without potential (fluctuations of internal

fields)• Problems: long range forces + loss of predictive power• Moduli stabilization

Flux compactifications (90s)

I Enlarge landscape: fluxes of antisymmetric tensor fieldsI Generate potential for "moduli" fields• Special holonomy → manifolds with G -structure• Scalar potential + extended SUSY: gauged supergravity• Fluxes ↔ gauging parameters

Scherk–Schwarz reductions - twisted tori - geometric fluxes

I Twisted reduction on a torus: twisted torus[Scherk, Schwarz ’79; Kaloper, Myers ’99]

• Induces scalar potential and non-abelian gauge symmetry• Consistent truncation on a (local) group manifold

Are there more general backgrounds/fluxes ?

I Non-geometric backgrounds: T-folds[Flournoy, Wecht, Williams ’02; Hellerman, McGreevy, Williams ’02; Dabholkar, Hull ’02;

Kachru, Schulz, Tripathy, Trivedi ’02]

• Background data: patched with T-dualities• Associated non-geometric fluxes (from duality covariance)

[Shelton, Taylor, Wecht ’05; Dabholkar, Hull ’05; Aldazabal, Camara, Font, Ibanez ’06; Shelton,

Taylor, Wecht ’06]

Are there more general effective theories ?

I General framework: embedding tensor[de Wit, Samtleben, Trigiante ’05]

• Yields all supergravity gaugings for N ≥ 1

Non-geometric backgrounds and doubled geometries

I Inherently stringy constructions: winding modes• Doubling of coordinates

[Witten ’88; Duff ’90; Tseytlin ’90, ’91; Kugo, Zwiebach ’92; Maharana, Schwarz ’92; Siegel ’93]

• Doubled torus: geometry for non-geometric string backgrounds[Hull ’04, ’06, ’07]

• Twisted doubled torus (TDT): electric N = 4 gaugings[Hull, Reid-Edwards ’07; Dall’Agata, Prezas, Samtleben, Trigiante ’07]

Worldsheet approach: chiral bosons

I Momentum and winding modes on equal footing: chiral bosons[Siegel ’84; Floreanini, Jackiw ’87]

• Interacting models: conditions from 2d Lorentz invariance[Tseytlin ’90, ’91]

• TDT as target space: Lorentz invariant![Dall’Agata, Prezas ’08]

• Conditions for conformal invariance[Avramis, Derendinger, Prezas ’09]

• Computation of the one-loop effective action• Agreement with corresponding gauged supegravity

Motivation and overview

Fluxes, twists and supergravity gaugings

Non-geometric backgrounds and twisted doubled tori

Interacting chiral boson models: Lorentz invariance

Interacting chiral boson models: one-loop effective action

Summary and open problems

Compactifications without fluxes

I Consider a toroidal reduction of string/M-theory• Abelian gauge theory without scalar potential• Existence of global duality symmetries

[Cremmer, Scherk, Ferrara ’79; Cremmer, Julia ’79]

Flux compactifications

I Our setup: heterotic strings on T6 (ignoring the gauge sector)I Let us assume NS-NS 3-form flux in the internal torus:

〈Hijk(x , y)〉 = hijk

• Decompose 2-form, metric tensor and dilaton:

BMN(x , y) → Bµν(x),Bµi (x),Bij (x)GMN(x , y) → Gµν(x),Gµi (x),Gij (x)

φ(x , y) → φ(x)

• 3-form kinetic term yields potential for metric moduli∫M4×T6

H ∧∗H =∫M4

d4x√−g4(hijkhmlnG im(x)G jl (x)G kn(x)+ · · · )

and non-abelian gauge interactions∫M4

d4x√−g4(· · ·+ ∂µB i

νGµjG νkhijk + · · · )

• Generators: Zi for Gµi , X i for Bµi . Gauge algebra:

[X i ,X i ] = 0[Zi ,X j ] = 0[Zi ,Zj ] = hijkX k

Scherk-Schwarz reductions (a.k.a. twisted tori, geometric fluxes)

I Can we enrich the previous gauge algebra ?I Consider reduction ansatz

ds2(10) = Gµν(x)dxµdxν

+ (δab + Gab(x))(ηa + Aaµ(x)dx

µ)(ηb + Abν(x)dx

ν)

with ηa(y) being left-invariant 1-forms satisfying

dηa = −12

τ abc ηb ∧ ηc

• Similar expansion for other fields• τ a

bc are structure constants of Lie algebra g (Jacobi identity)I Consistent truncation to singlets under GL

[Duff, Pope ’85; Hull, Reid-Edwards ’05]

I Geometric flux: spin connection condensate

〈ω abc (x , y)〉 = 1

2τ abc

• Compactification on a twisted torus: local group manifold G/ΓI Equivalent to a twisted reduction on a torus

[Scherk, Schwarz ’79]

ηa(y) = Uaj (y)dy

i

• Introduces y -dependance that is eliminated if

U−1ib (y)U−1j

c (y)∂[iUaj ] = τ a

bc

I Combining with NS-NS flux yields gauge algebra:[Kaloper, Myers ’99]

[X a,X b] = 0[X a,Zb] = τ a

bc X c

[Za,Zb] = τ cab Zc + habcX c

• Part of isometry group GR appears as gauge symmetry

• Introduce XA = Za,X a, gauge algebra:

[XA,XB ] = TABCXC

• Fluxes ⇒ gauge algebra structure constants TABC

• Jacobi for TABC : Bianchi for fluxes and Jacobi for τ abc

I Pack scalar fields in O(6,6)O(6)×O(6) matrix:

MAB =

(Gab − BacG cdBdb BacG cb

−GbcBca G ab

)• Scalar potential:

V ∼ T CDA T D

CB MAB +13T ACE T B

DF MABMCDMEF

I What type of theory is this ?• 4 dimensions, 16 supersymmetries, gravity, non-abelian gauge

symmetry, scalar potential =⇒ N = 4,D = 4 gaugedsupergravity[Kaloper, Myers ’99]

General gauge algebras[Shelton, Taylor, Wecht ’05; Dabholkar, Hull ’05; Aldazabal, Camara, Font, Ibanez ’06]

I Fill in the gaps in Kaloper–Myers gauge algebra:

[Za,Zb] = τ cab Zc + habcX c

[X a,Zb] = τ abc X c −Q ac

b Zc[X a,X b] = Q ab

c X c + RabcZc

• Physical and geometric fluxes: non-semisimple gauge groups• Q ab

c : non-abelian gauge symmetry from B-field gauge vectors!I What is the origin of the new "fluxes" Q and R ? T-dualityI Organizing principle: N = 4,D = 4 gauged supergravity

Gauged supergravity perspective

I General gauging: adjoint of gauge group → fundamental ofduality group[de Wit, Samtleben, Trigiante ’05]

XA = θmA tm

• Embedding tensor θaM : subject to conditions ⇒ full theory• In particular: gauge algebra ⇒ all possible fluxes!

I Focus on N = 4,D = 4 gauged supergravity• N = 4 graviton multiplet: graviton, axio-dilaton, 6 vectors• N = 4 vector multiplet: vector, 6 scalars• Duality group of ungauged theory: SL(2)×O(6, 6+ n)• Scalars

τ ∈ SL(2)U(1)

, MAB ∈O(6, 6+ n)

O(6)×O(6+ n)

• Embedding tensor (subject to quadratic constraints)[Schön, Weidner ’06]

fαABC , ξαA

• Heterotic reduction (n = 0): T-duality group O(d , d), d = 6

• Electric gaugings

f−ABC = 0, ξαA = 0

• For electric gaugings: embedding tensor ≡ structure constants

f+ABC := fABC = TABDηCD = TABC

• O(d , d) metric ηAB

ηAB =

(0 1d

1d 0

)• N = 4 gauge algebra

[XA,XB ] = TABCXC

• ηAB invariant metric on gauge algebra• N = 4 scalar potential

V (M) ∼ 12

(13MAA′MBB ′MCC ′ +

(23

ηAA′ −MAA′)

ηBB ′ηCC ′)TABCTA′B ′C ′

• Consider geometric T6 symmetry: GL(6) ⊂ O(6, 6)I Decompose fundamental rep 12→ 6⊕ 6

XA = Za,X a

fABC = (fabc , f cab , f bc

a , f abc)

• N = 4 electric gaugings: include all possible (NS-NS) fluxes[Derendinger, Petropoulos, Prezas ’07]

habc , τ cab ,Q bc

a ,Rabc

• There is a similar set of S-dual fluxes: f−ABC• ξαI : reduction with SO(1, 1) duality twist

[Derendinger, Petropoulos, Prezas ’07]

I Similar story: N = 8 gauged supergravity (type II/M-theory )

Motivation and overview

Fluxes, twists and supergravity gaugings

Non-geometric backgrounds and twisted doubled tori

Interacting chiral boson models: Lorentz invariance

Interacting chiral boson models: one-loop effective action

Summary and open problems

Duality related backgrounds

I h, τ,Q,R are related by T-dualityI Toy model (not a solution but can be promoted to one)

[Kachru, Schulz, Tripathy, Trivedi ’02; Lowe, Nastase, Ramgoolam ’03]

• 3-torus with N units of NS-NS flux hxyz

ds2 = dx2 +dy2 +dz2, H = Ndx ∧dy ∧dz , B = Nxdy ∧dz

• x ∼ x + 1: gauge transformation on BI T-duality along z ⇒ twisted torus (geometric flux τz

xy )

ds2 = dx2 + dy2 + (dz +Nxdy)2, H = 0

τy ,z (x) = Nx + i

• x ∼ x + 1: SL(2;Z) modular transformation on fiber T2y ,z

• Bianchi II or nilmanifold

I T-duality along y ⇒ T-fold (Q-flux Qzyx )

ds2 = dx2 +1

1+N2x2 (dz2 + dy2), B =

Nx1+N2x2 dy ∧ dz

• x ∼ x + 1: T-duality group O(2, 2;Z) action onMAB ≡ Gab,Bab

MAB(x + 1) =

1 0 0 00 1 0 00 −N 1 0N 0 0 1

MAB(x)

• There is a local geometric description• Globally well-defined up to T-duality• In this sense: non-geometric

I T-duality along x : cannot be done using Buscher rules• At the level of the effective theory: R-flux Rxyz

• In this case there is no local geometric description• Explicit dependance on dual coordinate xI Other non-geometric string constructions: S-folds, U-folds,

mirror-folds, asymmetric orbifolds, CFT constructions

Doubling and doubled tori[Witten ’88; Duff ’90; Tseytlin ’90, ’91; Kugo, Zwiebach ’92; Maharana, Schwarz ’92]

I y i conjugate to momenta, y i conjugate to windings• Double coordinates Y I = y i , y i ⇒ doubled torus T2n

dS2 = 2dy idyi , O(n, n;Z) ∈ GL(2n;Z)

• Realize T-duality group O(n, n;Z) geometricallyI Geometry for non-geometric string backgrounds

[Hull ’04, ’06, ’07]

I Scherk–Schwarz: torus → twisted torus → geometric fluxesI Natural idea: why not double everything and then twist• Generalization of Scherk–Schwarz on the doubled torus!

Twisted doubled tori (TDT)[Hull, Reid-Edwards ’07; Dall’Agata, Prezas, Samtleben, Trigiante ’07]

I Consider ordinary compactification on Td

• Full gauge group U(1)2d : isometry of the doubled torus T2d

• Suggests: generic gauging G from twisting the doubled torus!• Generalization of Scherk–Schwarz via a double field theoryI Gauge algebra and Gab potential in Scherk-Schwarz reduction

(twisted torus) of pure gravity:

[Za,Zb] = τkabZc

V ∼ 2 τabc τb

ad gcd + τa

bc τdef gadg

beg cf

• Gauge algebra XA = Za,X a and MAB ≡ Gab,Babpotential for electric N = 4 gaugings (twisted doubled torus)

[XA,XB ] = TABCXC

V ∼ 3 T CDA T D

CB MAB + T ACE T B

DF MABMCDMEF

• MAB "metric" moduli on twisted doubled torus

I General gauging: Scherk–Schwarz – but doubled!I Geometric flux on doubled torus ⇔ all fluxesI Twisted doubled torus (TDT): local group manifold G

EA = EAI (Y )dY I

dEA = −12T ABCE

B ∧ EC

T ABC = habc , τ a

bc ,Q bca ,Rabc

I Geometric interpretation of embedding tensorI Gauging ⇔ Fluxes ⇔ TDT =⇒ spacetime data (partially)

[Dall’Agata, Prezas, Samtleben, Trigiante ’07]

I Is it really the underlying geometry or just bookkeeping tool ?

Motivation and overview

Fluxes, twists and supergravity gaugings

Non-geometric backgrounds and twisted doubled tori

Interacting chiral boson models: Lorentz invariance

Interacting chiral boson models: one-loop effective action

Summary and open problems

Chiral bosons

I Twisted doubled torus as string target space ?• Treat momentum and winding independentlyI Chiral boson theory

[Siegel ’84; Floreanini, Jackiw ’87]

• Floreanini–Jackiw (FJ) approach (∂± = ∂0 ± ∂1)

S+ =12

∫d2σ∂1φ+∂−φ+

∂1(∂−φ+) = 0⇒ ∂−φ+ = 0

• Lorentz invariant on-shell

δLS+ =12

∫d2σ(∂−φ+)

2

• Similarly for antichiral boson S− = − 12

∫d2σ∂1φ−∂+φ−

I Introducing usual and dual fields

φ =1√2(φ+ + φ−), χ =

1√2(φ+ − φ−)

S = S++S− =12

∫d2σ

(∂0φ∂1χ+ ∂1φ∂0χ− (∂1φ)2− (∂1χ)2

)• Symmetric under φ↔ χ

• Double integration by parts, set p = ∂1χ: first order

S =12

∫d2σ

(2∂0φp − (∂1φ)2 − p2

)• Non-local duality relation p ≡ ∂1χ = ∂0φ

• Second order action of free boson φ

S =12

∫d2σ

((∂0φ)2 − (∂1φ)2

)

I T-duality invariant formulation• Start with compact boson at radius R

S =R2

2

∫d2σ

((∂0φ)2 − (∂1φ)2

)• First order

S =12

∫d2σ

(− p2

R2 + 2p∂0φ− (R∂1φ)2)

• Introduce dual field ∂1χ = p, integrate by parts

S =12

∫d2σ

(∂1χ∂0φ + ∂0χ∂1φ− (R−1∂1χ)2 − (R∂1φ)2

)• Manifest T-duality invariance: φ↔ χ,R ↔ R−1

• Dual boson at radius R−1

S =R−2

2

∫d2σ

((∂0χ)2 − (∂1χ)2

)

Interacting chiral bosons

I Consider the following action (YI = y i , y i)[Tseytlin ’90, ’91]

S =12

∫d2σ

(−HIJ (Y)∂1YI ∂1YJ +

(ηIJ (Y) + CIJ (Y)

)∂0YI ∂1YJ

)• HIJ , ηIJ : symmetric, CIJ antisymmetric• Interacting generalization of Floreanini-JackiwI Free theory

HIJ =

(1d 00 1d

), ηIJ =

(0 1d

1d 0

), CIJ = const.

• Yields d + d copies of FJ for y i± = y i ± y i : double torus• Integrate out y i (or y i ): original (or dual) torus vacuum

I Lorentz invariance condition

ηIJ

(∂0YI ∂0YJ + ∂1YI ∂1YJ

)− 2HIJ∂0YI ∂1YJ = 0

• Rearrange to(η −Hη−1H

)IJ ∂1YI ∂1YJ + ηIJVIVJ = 0 with

VI ≡ ηIJ∂0YJ −HIJ∂1YJ

• Sufficient (η −Hη−1H

)IJ = 0 and ηIJVIVJ = 0

• Notice: equation of motion

2∂1VI + ∂IHJK ∂1YJ∂1YK

− GIJK ∂0YJ∂1YK − 2ηJLΓLIK (η)∂0YJ∂1YK = 0

I Lorentz invariant modelsI Standard sigma models

[Tseytlin ’90, ’91]

HIJ =

(Gij − BikGklBlj BikGkj

−G ikBkj G ij

), ηIJ =

(0 1d

1d 0

),CIJ = const.

• Metric Gij = Gij (y), B-field Bij = Bij (y)• Integrating out y i

S =∫

d2σ(√

γγµνGij + εµνBij )∂µy i∂νy j

I Interacting non-abelian chiral scalars[Depireux, Gates, Park ’89; Gates, Siegel ’88; Tseytlin ’90, ’91]

I η = ±H: quantum Lorentz anomaly[Dall’Agata, Prezas ’08]

I Twisted doubled tori[Dall’Agata, Prezas ’08]

• Gauge algebra → group representative → vielbein

dEA = −12T ABCE

B ∧ EC , EA = EAI (Y )dY I

• Construct ηIJ and HIJ as

ηIJ = ηABEAI E

BJ ; ηAB =

(0 1d

1d 0

)HIJ = HABEA

I EBJ ; ηAB = HACηCDHDB

• HAB ∈ O(d ,d)O(d)×O(d) (cf. N = 4 moduli MAB)

• First Lorentz invariance condition is satisfied by construction• Add generalized three-form flux

GABC = 3∂[ACBC ] = TABC

• Equation of motion reads

(DIVJ −12T KIJ VK )∂1YI = 0

• Second Lorentz invariance condition is satisfied• Compact gaugings

TABDHDC = −TACDHDB

• Choose GABC = −TABC ⇒ V I = 0• Electric N = 4 gauging and moduli matrix MAB ⇔

Lorentz invariant chiral boson model• Related (equivalent ?) second order constrained formulation

[Albertsson, Kimura, Reid-Edwards ’08, Hull, Reid-Edwards ’09, Reid-Edwards ’09]

I Application: recover spacetime background fields[Dall’Agata, Prezas ’08]

• Impossible to integrate out dual coordinates for R-flux• No local geometric description

Motivation and overview

Fluxes, twists and supergravity gaugings

Non-geometric backgrounds and twisted doubled tori

Interacting chiral boson models: Lorentz invariance

Interacting chiral boson models: one-loop effective action

Summary and open problems

One-loop effective action

I Can the TDT models be promoted to conformal field theories ?I Starting point

S =12

∫d2σ

(−HIJ (Y)∂1YI ∂1YJ +

(ηIJ (Y) + CIJ (Y)

)∂0YI ∂1YJ

)• Earlier work: HIJ and ηIJ depend on base coordinates Xm

[Berman, Copland, Thompson ’07; Berman, Copland ’07]

• Background field method YI = YIcl + πI (ξI )

[Alvarez-Gaume, Freedman, Mukhi ’81; Braaten, Curtright, Zachos ’85; Mukhi ’86; Howe,

Papadopoulos, Stelle ’88]

• Covariant expansion, recursive algorithm[Mukhi ’86]

Sn =1n!DnS ≡ 1

n!

(∫d2σξI (σ)

DDYI (σ)

)n

S

I η-covariant expansion

S1 =∫

d2σ( 1

2ηIJ (∂0YID1ξJ +D0ξI ∂1YJ )−HIJ∂1YID1ξJ

− 12DKHIJ ξK ∂1YI ∂1YJ +

12GIJK ξK ∂0YI ∂1YJ

)

S2 =12

∫d2σ

(−HIJD1ξID1ξJ + ηIJD0ξID1ξJ

+

(12DJGIKL + RKIJL

)ξI ξJ∂0YK ∂1YL

− 12(DIDJHKL +HKMRM

IJL +HLMRMIJK )ξI ξJ∂1YK ∂1YL

+12GIJK ξK (∂0YID1ξJ +D0ξI ∂1YJ )− 2DKHIJξKD1ξI ∂1YJ

)

I One-loop effective action Seff = Scl + Γ

exp (iΓ[Y]) =∫Dξ exp (iS2[Y; ξ])

I Propagators of tangent space fields ξA

[Tseytlin ’90, ’91; Berman, Copland, Thompson ’07]

〈ξA(σ)ξB(σ′)〉 = HAB∆(σ− σ′) + ηAB ∆(σ− σ′)

∆(σ− σ′) =12(∆+(σ− σ′) + ∆−(σ− σ′)

)= − 1

4πln(σ− σ′)2

∆(σ− σ′) =12(∆+(σ− σ′)− ∆−(σ− σ′)

)= − 1

2πarctanh

σ1 − σ′1

σ0 − σ′0

• Regularize: ln(σ− σ′)2 → ln((σ− σ′)2 + µ2), set

σ1−σ′1

σ0−σ′0 = tanh δ

• Limiting expressions

∆(0)→ − 12π

ln µ , ∆(0)→ − 12π

δ

• Breakdown of scale invariance and Lorentz invariance

I Weyl anomaly and global Lorentz anomaly

W =12

∫d2σ

(W 00

IJ ∂0YI ∂0YJ +W 01IJ ∂0YI ∂1YJ +W 11

IJ ∂1YI ∂1YJ)

L =12

∫d2σ

(L00IJ ∂0YI ∂0YJ + L01IJ ∂0YI ∂1YJ + L11IJ ∂1YI ∂1YJ

)• Vanishing on-shell: equations of motion and/or constraints• Weyl anomaly coefficients:

W 00IJ =

14(HA[CHD]B − ηA[C ηD]B )Q0

AB,IQ0CD,J

W 01IJ = HABS01AB,IJ +

12(HA[CHD]B − ηA[C ηD]B )Q0

AB,IQ1CD,J

− 14(HA[C ηD]B + ηA[CHD]B )

(Q0

AB,IP1CD,J + P1

AB,IQ0CD,J

)W 11

IJ = HABS11AB,IJ +14(HA[CHD]B − ηA[C ηD]B )Q1

AB,IQ1CD,J

− 14(HA[C ηD]B + ηA[CHD]B )

(Q1

AB,IP1CD,J + P1

AB,IQ1CD,J

)− 1

4(HA[CHD]B + 3ηA[C ηD]B )P1

AB,IP1CD,J

• Lorentz anomaly coefficients

L00IJ = 0

L01IJ = ηABS01AB,IJ −12

ηA[C ηD]B (Q0AB,IP1

CD,J + P1AB,IQ0

CD,J)

L11IJ = ηABS11AB,IJ −12(HA[C ηD]B + ηA[CHD]B )P1

AB,IP1CD,J

− 12

ηA[C ηD]B (Q1AB,IP1

CD,J + P1AB,IQ1

CD,J)

• Basic building blocks:

S11AB,IJ = − 12DADBHIJ −

12(HIKRK

ABJ +HJKRKABI )−HCDΩ C

I AΩ DJ B

− DAHCI Ω CJ B −DAHCJΩ C

I B

S01AB,IJ = RIABJ +12DBGAIJ + ηCDΩ C

I AΩ DJ B +

12(GIDAΩ D

J B −GJDAΩ DI B )

Q1AB,I = −2Ω C

I AHCB − 2DAHBI

Q0AB,I = − 1

2GIAB + Ω C

I AηCB

P1AB,I =

12GIAB + Ω C

I AηCB

Specialize to twisted doubled tori

I Use general flux GABC = λTABCI Lorentz anomaly (TABC = T D

AB HDC ,TABI = T CAB HCDED

I )

L01IJ =

λ2 − 14TIABT AB

J

L11IJ = − (λ− 1)2

4TI ABT

ABJ − λ− 1

4(TI ABT

ABJ + TJABT

ABI )

• Vanish for all gaugings when λ = 1• Compact gaugings

L11IJ = −λ2 − 1

4TIABT

ABJ ,

• Vanish for λ = ±1 as well• Classical Lorentz invariance, once established, survives inthe quantum theory

I Weyl anomaly

W 00IJ =

(λ− 1)2

16(TIABT AB

J − TIABT ABJ )

W 01IJ =

λ− 14

(TIABT AB

J − TIABT ABJ + (λ + 1)TIABT AB

J

)W 11

IJ =(λ + 1) (8− 3(λ + 1))

16TIABT AB

J +(λ + 1) (8− (λ + 1))− 16

16TIABT AB

J

− 14(TIABT

ABJ − TIABT

ABJ )− λ− 1

4(TIABT

ABJ + TJABT

ABI )

• General gaugings with λ = 1: W 00IJ = W 01

IJ = 0• Condition for (one-loop) conformal invariance

W 11AB =

14(ηAE ηBE ′ −HAEHBE ′ )(η

CC ′ηDD ′ −HCC ′HDD ′ )T ECD T E ′

C ′D ′ = 0

• Compact gaugings: W 00IJ = W 01

IJ = 0 for λ = ±1 and

W 11IJ = −λ2 − 1

4TIABT AB

J

• Conformal models for λ = ±1

Correspondence with N = 4 electric gaugings

I Introduce projectors (recall η = Hη−1H)

PABCD± (H) =

12(ηA(CηD)B ±HA(CHD)B)

• Define

ZAB(H) =12(ηC [C ′ηD ′]D −HC [C ′HD ′]D)TCDATC ′D ′B

• Weyl anomaly

WAB = P CD−AB (H)ZCD(H)

• N = 4 gauged supergravity potential (electric gaugings)

V (M) =12

(13MAA′MBB ′MCC ′ +

(23

ηAA′ −MAA′)

ηBB ′ηCC ′)TABCTA′B ′C ′

• Incorporate constraint MAB ∈ O(d ,d)O(d)×O(d)

V = V + Λ BA (MACηCB − ηACMCB)

• Condition for extremum: set to zero

∂V∂MAB = −ZAB(M) +

12(ηACηBD +MABMCD)ΛCD

= −ZAB(M) + P+ABCD(M)ΛCD

• Identify HAB = MAB and act with P−(H)

PABCD− (H)ZCD(H) = 0

• N = 4 electric vacuum ⇒ conformal chiral boson model

I Assume now conformal invariance condition is satisfied. Then

ZAB(H) = P+ABCD(H)ZCD(H)

• Thus

∂V∂MAB

∣∣∣∣M=H

= P+ABCD(H)(

ΛCD − ZCD(H))

• N = 4 vacuum for ΛCD = ZCD(H)

I N = 4 electric vacua ⇔ conformal chiral boson modelsI SUSY: TDT is even-dim group manifold ⇒ N = 2 SCFT

Compact gaugings and chiral WZW models

I Compact gaugings ⇒ conformal chiral boson modelsI Compact gauging conditions for HAB = δAB

habc = Q bca , Rabc = τ a

bc , τ cab = −τ b

ac , Q abc = −Q cb

a

• Change basis Za,X a to T±a = 12 (Za ± X a)

• Gauge algebra is a sum

[T±a ,T±b ] = f ±cab T±c , [T+a ,T−b ] = 0

f ±cab = τ cab ±Q bc

a

• Gauge group GL × GR ∈ O(d)×O(d)

I Corresponding chiral boson model (chiral basis y iL,R = y i ± y i )

S =14

∫d2σ

((GL

ij + λBLij )∂−y

iL∂1y

jL − (GR

ij + λBRij )∂+y

iR∂1y

jR

)• GL,R

IJ and dBL,R invariant metric and 3-form of GL,R• Chiral and antichiral WZW based on GL and GR

[Sonnenscheim ’88; Tseytlin ’91]

• Kac–Moody symmetry ⇒ all order conformal invariance!• When GL = GR : ordinary WZW model• In line with previous observations for GL = GR = SU(2)

[Dabholkar, Hull 05; Dall’Agata, Prezas ’08]

• WZW model is realized in terms of non-geometric fluxes• Notion of non-geometricity needs refinement

Motivation and overview

Fluxes, twists and supergravity gaugings

Non-geometric backgrounds and twisted doubled tori

Interacting chiral boson models: Lorentz invariance

Interacting chiral boson models: one-loop effective action

Summary and open problems

Summary

I Supergravity gaugings: non-geometric string backgrounds• Embedding tensor ⇒ gauge algebra ⇒ all fluxes!• New interpretation: geometric flux on doubled torus• Twisted doubled tori: underlying geometriesI Interacting chiral bosons: doubled geometry sigma models• Twisted doubled tori ⇒ Lorentz invariant chiral bosons• Conformal invariance condition ⇔ Vacua of N = 4 potential• Compact gaugings are always conformal: chiral WZW modelsI Twisted doubled tori are actual physical backgroundsI Scherk–Schwarz on doubled geometries → general

supergravity gaugings

Open problems

I Straightforward generalizations• Include dilaton (gaugings with SO(1, 1) duality twist)• Include non-compact spacetime dimensions• Include worldsheet supersymmetry• Full equations of motion of N = 4 gauged supergravityI Other classes of Lorentz invariant chiral boson models ?I Conformal field theory aspects• Origin of conformal invariance condition• Deformations of (chiral, gauged) WZW models ?

[Tseytlin ’93]

• Relation to affine Virasoro constructions

I Relation to other worldsheet models[Albertsson, Kimura, Reid-Edwards ’08; Hull, Reid-Edwards ’09; Reid-Edwards ’09; Halmagyi ’09]

• Non-commutativity of Q-flux and non-associativity of R-flux ?[Mathai, Rosenberg ’04; Bouwknegt, Hannabuss, Mathai ’04; Ellwood, Hashimoto ’06]

I Connection with double field theory[Hohm, Hull, Zwiebach ’10]

• Equations of motion: beta functionals• Explicit Scherk–Schwarz on twisted doubled torus: electricN = 4 gaugings

I Exotic constructions• Incorporate SL(2,Z) S-duality, double the doubled torus!• Account for all N = 4 gaugings• E7 invariant membrane theory!• Account for all N = 8 gaugings